We introduce a new family of quantum circuits for which the scrambling of a subspace of nonlocal operators is classically simulable. We call these circuits "super-Clifford circuits" since the Heisenberg time evolution of these operators corresponds to Clifford evolution in operator space. Thus we are able to classically simulate the time evolution of certain single Pauli strings into operators with operator entanglement that grows linearly with the number of qubits. These circuits provide a new technique for studying scrambling in systems with a large number of qubits, and are an explicit counter example to the intuition that classical simulability implies the absence of scrambling.
We introduce a new family of quantum circuits for which the scrambling of a subspace of nonlocal operators is classically simulable. We call these circuits "super-Clifford circuits" since the Heisenberg time evolution of these operators corresponds to Clifford evolution in operator space. Thus we are able to classically simulate the time evolution of certain single Pauli strings into operators with operator entanglement that grows linearly with the number of qubits. These circuits provide a new technique for studying scrambling in systems with a large number of qubits, and are an explicit counter example to the intuition that classical simulability implies the absence of scrambling.
The complexity of simulating quantum many-body dynamics, or quantum computations, in the Heisenberg picture is governed by the scrambling of initially simple operators into superpositions of exponentially many Pauli strings. The corresponding expansion coefficients define the Pauli spectrum, whose structure controls the performance of classical algorithms based on truncating Pauli expansions. Here we determine the finite-depth Pauli spectrum of random quantum circuits, both in the noiseless case and in the presence of local noise, through its moments, given by the operator stabilizer Rényi entropies. In noiseless circuits, we uncover a hierarchy in the approach to the fully scrambled regime: low moments equilibrate at relatively short depths, while higher moments, which are sensitive to rare, large-amplitude Pauli coefficients, require parametrically larger depths. In noisy circuits, scrambling competes with an effective suppression of operator spreading. Above a critical error per cycle γcN=O(1), the operator fails to reach the fully scrambled distribution and remains supported on an atypically sparse subset of Pauli strings. Conversely, below this scale, we rigorously show that classical simulation remains exponentially hard, demonstrating that finite noise does not automatically imply classical simulability. The resulting noise-induced transition in operator complexity therefore delineates the boundary between intrinsically hard quantum dynamics and those that remain classically accessible.
We study operator scrambling in quantum circuits built from “super-Clifford” gates. For such circuits it was established by Blake and Linden [] that the time evolution of operator entanglement for a large class of many-body operators can be efficiently simulated on a classical computer, including for operators with volume-law entanglement. Here we extend the scope of this formalism in two key ways. First we provide evidence that these classically simulable circuits include examples of fast scramblers, by constructing a circuit for which operator entanglement is numerically found to saturate in a time t * ∼ ln ( N ) (with N the number of qubits). Second we demonstrate that, in addition to operator entanglement, certain out-of-time-ordered correlation functions (OTOCs) can be classically simulated within the same formalism. As a consequence such OTOCs can be computed numerically in super-Clifford circuits with thousands of qubits and we study several explicit examples in the aforementioned fast scrambling circuits.
Random permutation circuits were recently introduced as minimal models for local many-body dynamics that can be interpreted both as classical and quantum. Standard dynamical complexity indicators such as damage spreading and out-of-time-order correlators (OTOCs), show that these systems exhibit sensitivity to initial conditions in the classical setting and operator scrambling in the quantum setting. Here, we address their quantum chaoticity—a stricter property—by studying the time evolution of local operator entanglement (LOE). We show that the behavior of LOE in random permutation circuits depends on the dimension of the local configuration space q . When q = 2 , i.e., the circuits act on qubits, random permutations are Clifford and the LOE of any local operator is bounded by a constant, indicating that they are not truly chaotic. On the other hand, when the dimension of the local configuration space exceeds two, the LOE grows linearly in time. We prove this in the limit of large q and present numerical evidence that a three-dimensional local configuration space is sufficient for a linear growth of LOE. Our findings highlight that quantum chaos can be produced by essentially classical dynamics. Moreover, we show that LOE can be defined also in the classical realm and put it forward as a universal indicator chaos, both quantum and classical.
Operator scrambling is a crucial ingredient of quantum chaos. Specifically, in the quantum chaotic system, a simple operator can become increasingly complicated under unitary time evolution. This can be diagnosed by various measures such as square of the commutator (out-of-time-ordered correlator), operator entanglement entropy etc. In this paper, we discuss operator scrambling in three representative models: a chaotic spin-1/2 chain with spatially local interactions, a 2-local spin model and the quantum linear map. In the first two examples, although the speeds of scrambling are quite different, a simple Pauli spin operator can eventually approach a "highly entangled" operator with operator entanglement entropy taking a volume law value (close to the Page value). Meanwhile, the spectrum of the operator reduced density matrix develops a universal spectral correlation which can be characterized by the Wishart random matrix ensemble. In the second example, we further connect the 2-local model into a one dimensional chain and briefly discuss the operator scrambling there. In contrast, in the quantum linear map, although the square of commutator can increase exponentially with time, a simple operator does not scramble but performs chaotic motion in the operator basis space determined by the classical linear map. We show that once we modify the quantum linear map such that operator can mix in the operator basis, the operator entanglement entropy can grow and eventually saturate to its Page value, thus making it a truly quantum chaotic model.
How much do local and time-periodic dynamics resemble a random unitary? In the present work, we address this question by using the Clifford formalism from quantum computation. We analyze a Floquet model with disorder, characterized by a family of local, time-periodic, and random quantum circuits in one spatial dimension. We observe that the evolution operator enjoys an extra symmetry at times that are a half-integer multiple of the period. With this, we prove that after the scrambling time, namely, when any initial perturbation has propagated throughout the system, the evolution operator cannot be distinguished from a (Haar) random unitary when all qubits are measured with Pauli operators. This indistinguishability decreases as time goes on, which is in high contrast to the more studied case of (time-dependent) random circuits. We also prove that the evolution of Pauli operators displays a form of mixing. These results require the dimension of the local subsystem to be large. In the opposite regime, our system displays a novel form of localization, produced by the appearance of effective one-sided walls, which prevent perturbations from crossing the wall in one direction but not the other.
Extended Clifford circuits straddle the boundary between classical and quantum computational power. Whether such circuits are efficiently classically simulable seems to depend delicately on the ingredients of the circuits. While some combinations of ingredients lead to efficiently classically simulable circuits, other combinations, which might just be slightly different, lead to circuits which are likely not. We extend the results of Jozsa and Van den Nest [Quant. Info. Comput. 14, 633 (2014)] by studying two further extensions of Clifford circuits. First, we consider how the classical simulation complexity changes when we allow for more general measurements. Second, we investigate different notions of what it means to ‘classically simulate’ a quantum circuit. These further extensions give us 24 new combinations of ingredients compared to Jozsa and Van den Nest, and we give a complete classification of their classical simulation complexities. Our results provide more examples where seemingly modest changes to the ingredients of Clifford circuits lead to “large” changes in the classical simulation complexities of the circuits, and also include new examples of extended Clifford circuits that exhibit “quantum supremacy”, in the sense that it is not possible to efficiently classically sample from the output distributions of such circuits, unless the polynomial hierarchy collapses.